The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

Abstract

We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: $T((a_n))_n=1/(a_0+\cdots +a_n)$. We determine the corresponding measure $\mu$, which has an increasing and convex density on $\mathopen]0,1\mathclose[$, and we study some analytic functions related to it. The Mellin transform $F$ of $\mu$ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on $\mathopen]-1,\infty\mathclose[$ satisfying $F(0)=1$ and the functional equation $1/F(s)=1/F(s+1)-F(s+1)$, $s>-1$.